A lab setup uses a 1.5 meter long optical bench to hold a light that serves as the object, a lens, and a screen for the image. The bench has a ruler attached to it. For this lab, the object is at the 1.0 cm mark on an optical bench. The 25.0 cm focal length lens is at the 40.0 cm mark. The object is 8.0 cm tall. Where do you need to put the screen on the optical bench?

To find the position of the screen on the optical bench, we can use the lens equation:

1/f = 1/d_o + 1/d_i

where:
f is the focal length of the lens,
d_o is the object distance,
d_i is the image distance.

In this case, the object distance is given as 1.0 cm, and the focal length is given as 25.0 cm. We need to find the image distance.

Step 1: Convert the given values to meters for consistency:

Object distance (d_o) = 1.0 cm = 0.01 m
Focal length (f) = 25.0 cm = 0.25 m

Step 2: Substitute the given values into the lens equation:

1/0.25 = 1/0.01 + 1/d_i

Step 3: Solve for the image distance (d_i):

1.0/0.25 - 1/0.01 = 1/d_i

4 - 100 = 1/d_i

-96 = 1/d_i

d_i = -1/96 m

The negative sign indicates that the image is virtual and formed on the same side as the object. Since we are dealing with real measurements, the screen should be placed 1/96 m to the left (before) the lens, keeping in mind the object distance and lens position.

Step 4: Calculate the distance of the screen from the lens:

Distance from lens to screen = object distance - image distance

Distance from lens to screen = 0.01 m - (-1/96 m)

Distance from lens to screen = 0.01 m + 1/96 m

Distance from lens to screen = (0.01 * 96 + 1) / 96 m

Distance from lens to screen = (0.96 + 1) / 96 m

Distance from lens to screen = 1.96 / 96 m

Distance from lens to screen ≈ 0.0204 m (rounded to four decimal places)

Therefore, the screen should be placed approximately 0.0204 meters from the lens on the optical bench.

To solve this problem, we can use the lens equation:

1/f = 1/d_o + 1/d_i

where:
f is the focal length of the lens,
d_o is the distance from the object to the lens, and
d_i is the distance from the lens to the image.

First, we need to convert the given values to meters:

f = 25.0 cm = 0.25 m
d_o = 1.0 cm = 0.01 m

Now we can solve for d_i:

1/0.25 = 1/0.01 + 1/d_i

4 = 100 + 1/d_i

1/d_i = 4 - 100

1/d_i = -96

d_i = 1/(-96)

d_i ≈ -0.0104 m

Since distances cannot be negative in this context, the negative sign indicates that the image is formed on the same side as the object, which means it is a virtual image.

The distance from the lens to the image (d_i) is approximately -0.0104 m.

Now, to find the position of the screen, we need to calculate the distance from the object to the screen (d_scr):

d_scr = d_o + d_i

d_scr = 0.01 + (-0.0104)

d_scr ≈ -0.0004 m

Again, the negative sign indicates that the screen needs to be placed on the same side as the object, which means it should be positioned 0.0004 m before the object.

Therefore, you need to put the screen approximately 0.0004 meters before the 1.0 cm mark on the optical bench.